Qualitative and Quantitative Imaging in Electromagnetic Inverse Scattering Theory

Abstract

The inverse scattering problem is inherently nonlinear and improperly posed. Relevant study, such as the existence and uniqueness of the solution, the completeness of the far field pattern, etc., involves an abstruse mathematical theory. In our daily life, the inversion techniques play a significant role in areas such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing. This thesis is focused on the qualitative and quantitative reconstruction of shape and medium parameters of scattering objects in electromagnetic inverse scattering theory. The major contributions of this thesis are 1) the proposal of a novel cross-correlated error termand 2) the proposal of the sum-of-normregularized reconstruction algorithm. The significance of the former lies in the fact that the proposed error term fills up a gap hidden in the classical “state error Å data error” cost functional. In the optimization approaches, the data error term tends to recover the unknown properties of the objects directly from the measurement data, while the state error term attempts to ensure that the recovered results satisfy Maxwell’s equations in the field domain. In other words, the solution must behave well in both the measurement domain and the field domain. However, there is still a gap in between because the minor mismatch in the field domain is not monitored in the measurement domain. The proposed crosscorrelated error is a constraint which tends to get the mismatch in the field domain under control in the measurement domain. Therefore, one can say that this novel error term revolutionizes the formulation of the minimization functional of inversion techniques based on optimization theory. The significance of the latter is that the proposed reconstruction scheme enables us to excavate the joint information hidden in the formulation of multiple inverse source problems, without any significant additional computational effort. Although the sum-of-norm regularization is not necessarily the best regularization constraint for some complicated scatterers, it demonstrates at least two points: 1) for an inverse source problem, benefits can be obtained from use of different incident fields; 2) the sum-of-norm regularization brings better resolving ability due to the joint processing of the multiple contrast source vectors. The research results in this thesis are also applicable to the acoustic inverse scattering problems. Application of the qualitative and quantitative reconstruction approaches developed in this thesis to the experimental data in different areas of wave-field inversion would be very interesting as future work.